Prepositional phrase agreement with the rest of the sentence

Is the following sentence grammatically correct?

Our testers are actually very nice people with a soul.

Assuming it is grammatically correct, is it logically correct? Am I implying that all of our testers share a single soul?

I understand that there are less confusing alternatives, such as

Our testers are actually very nice people with souls.

... but I'm only concerned about the correctness of the first sentence.

Read literally, your first sentence does suggest that all the testers share a soul. But since the common understanding of the meaning of soul makes this unlikely, no one would expect that you actually meant it, so they would interpret it as your second sentence.

But if you replace soul with something that's likely to be shared, the inference would be taken. E.g.

Our testers are very nice people living in a house.

suggests that your testers are housemates.